Lemma 2.6 ([1]}) 
Let \(({{F}}_n)_{n \in \mathbb {N}}\)  be an increasing sequence of sub-\(\sigma \) -algebras of \({{F}}\) , let \((z_n)_{n \in \mathbb {N}},\ (\beta _n)_{n \in \mathbb {N}},\ (\theta _n)_{n \in \mathbb {N}}\)  and \((\gamma _n)_{n \in \mathbb {N}}\)  be \([0,+\infty ]\) -valued random sequences such that, for every \(n \in \mathbb {N}\) , \(z_n,\ \beta _n,\ \theta _n\)  and \(\gamma _n\)  are \({{F}}_n\) -measurable. Suppose that \(\sum _{n \in \mathbb {N}}\gamma _n<+\infty ,\ \sum _{n \in \mathbb {N}}\beta _n<+\infty \ a.s.\)  and
\((\forall n \in \mathbb {N})\ \mathsf {E}[z_{n+1}|{{F}}_n] \le (1+\gamma _n)z_n+\beta _n-\theta _n \ a.s..\) 
